Symmetry set

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An ellipse (red), its evolute (blue), and its symmetry set (green and yellow). the medial axis is just the green portion of the symmetry set. One bi-tangent circle is shown.

In geometry, the symmetry set is a method for representing the local symmetries of a curve, and can be used as a method for representing the shape of objects by finding the topological skeleton. The medial axis, a subset of the symmetry set is a set of curves which roughly run along the middle of an object.

The symmetry set in 2 dimensions[edit]

Let be an open interval, and be a parametrisation of a smooth plane curve.

The symmetry set of is defined to be the closure of the set of centres of circles tangent to the curve at at least two distinct points (bitangent circles).

The symmetry set will have endpoints corresponding to vertices of the curve. Such points will lie at cusp of the evolute. At such points the curve will have 4-point contact with the circle.

The symmetry set in n dimensions[edit]

For a smooth manifold of dimension in (clearly we need ). The symmetry set of the manifold is the closure of the centres of hyperspheres tangent to the manifold in at least two distinct places.

The symmetry set as a bifurcation set[edit]

Let be an open simply connected domain and . Let be a parametrisation of a smooth piece of manifold. We may define a parameter family of functions on the curve, namely

This family is called the family of distance squared functions. This is because for a fixed the value of is the square of the distance from to at

The symmetry set is then the bifurcation set of the family of distance squared functions. I.e. it is the set of such that has a repeated singularity for some

By a repeated singularity, we mean that the jacobian matrix is singular. Since we have a family of functions, this is equivalent to .

The symmetry set is then the set of such that there exist with , and

together with the limiting points of this set.

References[edit]

1. J. W. Bruce, P.J.Giblin and C. G. Gibson, Symmetry Sets. Proc. of the Royal Soc.of Edinburgh 101A (1985), 163-186.

2. J. W. Bruce and P.J.Giblin, Curves and Singularities, Cambridge University Press (1993).